Large BMO spaces vs interpolation

Abstract

We introduce a class of BMO spaces which interpolate with $L_p$ and are sufficiently large to serve as endpoints for new singular integral operators. More precisely, let $\left(\Omega ,\Sigma ,\mu \right)$ be a $\sigma$-finite measure space. Consider two filtrations of $\Sigma$ by successive refinement of two atomic $\sigma$-algebras ${\Sigma }_a$ and ${\Sigma }_b$ having trivial intersection. Construct the corresponding truncated martingale BMO spaces. Then, the intersection seminorm only leaves out constants and we provide a quite flexible condition on $\left({\Sigma }_a,{\Sigma }_b\right)$ so that the resulting space interpolates with $L_p$ in the expected way. In the presence of a metric $d$, we obtain endpoint estimates for Calderón–Zygmund operators on $\left(\Omega ,\mu ,d\right)$ under additional conditions on $\left({\Sigma }_a,{\Sigma }_b\right)$. These are weak forms of the “isoperimetric” and the “locally doubling” properties of Carbonaro, Mauceri and Meda which admit less concentration at the boundary. Examples of particular interest include densities of the form $e^{\pm |x{|}^{\alpha }}$ for any $\alpha >0$ or ${\left(1+|x{|}^{\beta }\right)}^{-1}$ for any $\beta \gtrsim n^{3∕2}$. A (limited) comparison with Tolsa’s RBMO is also possible. On the other hand, a more intrinsic formulation yields a Calderón–Zygmund theory adapted to regular filtrations over $\left({\Sigma }_a,{\Sigma }_b\right)$ without using a metric. This generalizes well-known estimates for perfect dyadic and Haar shift operators. In contrast to previous approaches, ours extends to matrix-valued functions (via recent results from noncommutative martingale theory) for which only limited results are known and no satisfactory nondoubling theory exists so far.